On well‐posedness of the first boundary‐value problem within linear isotropic Toupin–Mindlin strain gradient elasticity and constraints for elastic moduli
Victor A. Eremeyev
2023ZAMM ‐ Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik11 citationsDOIOpen Access PDF
Abstract
Abstract Within the linear Toupin–Mindlin strain gradient elasticity we discuss the well‐posedness of the first boundary‐value problem, that is, a boundary‐value problem with Dirichlet‐type boundary conditions on the whole boundary. For an isotropic material we formulate the necessary and sufficient conditions which guarantee existence and uniqueness of a weak solution. These conditions include strong ellipticity written in terms of higher‐order elastic moduli and two inequalities for the Lamé moduli. The conditions are less restrictive than those followed from the positive definiteness of the deformation energy.
Topics & Concepts
Mathematical analysisIsotropyMathematicsElasticity (physics)Boundary value problemUniquenessLinear elasticityModuliViscoelasticityPositive definitenessDirichlet boundary conditionPhysicsPositive-definite matrixEigenvalues and eigenvectorsFinite element methodThermodynamicsQuantum mechanicsNonlocal and gradient elasticity in micro/nano structuresThermoelastic and Magnetoelastic PhenomenaComposite Material Mechanics